Reactance Derived

Refer to Table 23.1, which shows the average cumulative effect of all the harmonics that may be present in a power system. If we can provide a series reactor of 6% of the total kVAr of the capacitor banks connected on the system, most of the harmonics present in the. system can be suppressed. With this reactance, the system would be tuned to below the fifth harmonic (at 204 Hz) for a 50 Hz system as derived below. 

TK solves the coupled ccpiations (11) for each J and E using the Jolmson-hfanolopoulos [8] logarithmic-derivative propagator. It determines also the reactance /v-matrix. 

By the same reasoning as for the transient reactances so the sub-transient reactances are derived, and are called the direct axis sub-transient reactance X and the quadrature axis subtransient reactance X" . 

The derived reactances were described in sub-section 3.2 in relation to their effect on armature reaction. They are derived from the actual winding reactances by the standard equations, for example References 8 and 9. 

Equation (7.2) can be used to calculate the situation at the time given for the breaking duty. Usually the sub-transient time decay term has fallen to zero, and the solution is in the transient period. When an external impedance is present its resistance can be included in the Ta time constant and its reactance added to the appropriate machine reactances. References 5 and 6 explain how the derived reactances and time constants are calculated and affected by the addition of the external impedance. 

Induction motors can be represented by the 2-axis theory, by using the derivations for synchronous machines but deleting the field winding. In this case some of the reactances become zero, and the field resistance is infinity. Hence, the derived reactances X ... 

In order to derive the familiar reactances e.g. Xj the sub-transient reactance, it is first necessary to obtain the operational impedances . (In control theory terminology these would be called transfer functions .)... 

The derived reactances are those most frequently used to specify synchronous generators and motors. They are the synchronous, transient and sub-transient reactances in the d and g-axes. The most convenient method of deriving these is from the application of a three-phase short circuit at the terminals of the unloaded machine, whether it be a generator or a motor. For a motor the testing procedure is more complicated as described in sub-section 5 of Reference 23. The factory tests. The g-axis are usually taken as their design values because the necessary factory tests are more difficult to perform. The tests are described in for example IEEE standard 112 and BS4296. 

In a situation where the disturbance is remote from the machine the short circuit time constants and the derived reactances X, X, X, Xq, (Xp, X" and X2 are all functions of the external reactance Xe since it should be added to Xa. Likewise Re should be added to Ra- Ra does not appear in the time constants except for Ta. 

Derived Reactances and Time Constants for an Induction Motor... 

The absence of the field winding can be used to convert the mathematical model of the synchronous machine into one for an induction machine. In addition the mutual inductance in the < -axis is made equal to mumal inductance in the d-axis, i.e. the machine becomes symmetrical in both axes. The matrix equations (20.6) to (20.16) are modified as shown below. In these equations the mutual inductances Mj and become M, Lim and Lihq become L/j., Rjut and R/aj become Rk. All the derived reactances and time constants for an induction machine are equivalent to those appUcable to the g-axis of the synchronous machine. 

It is reasonable to regard the rotor windings as damper windings and use the notation of sub-transient reactances. Hence the following derived reactances and time constants are appropriate to induction machines. 

Application of a three-phase short circuit to the terminals of an unloaded induction motor is not a practical factory test, especially for a large high-voltage motor, because the motor can only be excited at its stator windings from the power supply. A three-phase short circuit at or near the stator terminals can occur in practice e.g. damaged supply cable, damage in the cable terminal box. The parameters of the stator and rotor windings can be obtained from other factory tests. However, the derived reactance can be defined in the same manner as those for the synchronous machine, but with... 

Derived reactances and time constants for an induction motor... 

Figure 24.7. The proportional-plus-derivative controller. Derivative action is accomplished by a shunt capacitor C across Rf. When deviation from the setpoint is rapid, the low reactance of the capacitor causes less negative feedback—hence, greater amplifier gain. The derivative time resistor Ra allows adjustment of the magnitude of derivative control action to a given rate of change of the error signal. Courtesy of the Foxboro Company.

Mechanical Impedance of Muscle Tissue. The (input) mechanical impedance is the complex ratio between the dynamic force applied to the body and the velocity at the interface where vibration enters the body. The real and imaginaty parts of the mechanical impedance of human muscle in vivo are shown as a function of frequency in Fig. 10.3 (von Gierke et al., 1952). In this diagram the measured resistance (open circles) and reactance (diamonds) are compared with the predictions of a model, from which some tissue properties may be derived (see Table 10.1). It should be noted that... 

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